Order Arrival And Cancellation Intensities

Think of every price level in the book as having three taps feeding and draining it: new limit orders arriving (adding size), cancellations (removing size), and market orders eating through it (removing size, but from the front). The intensity of each is the average rate at which that type of event happens per unit time — not a single number for the whole book, but a rate that depends on where you are in the book and how much size is already sitting there.

The key empirical fact, and the reason this gets modelled at all: cancellation intensity rises with queue size. A level with 5,000 shares resting sees far more cancellations per second than one with 50, in absolute terms — more orders sitting there means more orders that might get pulled. This isn't obvious from first principles; it's an observed regularity that any realistic book model has to reproduce.

A worked example

Suppose at the best bid, historical data shows: new limit orders arrive at a rate of $\lambda = 8$ per second, cancellations occur at a rate of $\mu(Q) = 0.002 \times Q$ per second where $Q$ is the current queue size, and market-order executions hit the level at $\theta = 3$ per second. Right now $Q = 1{,}000$ shares.

The cancellation rate is $\mu(1000) = 0.002 \times 1000 = 2$ per second. The expected time until the next event of any kind (add, cancel, or execute) is the reciprocal of the total rate, $\lambda + \mu + \theta = 8 + 2 + 3 = 13$ per second, so about $1/13 \approx 0.077$ seconds — roughly 77 milliseconds between events at this level, on average. And the probability that the next event specifically is a cancellation (as opposed to an add or an execute) is $2/13 \approx 15.4\%$.

Now suppose the queue grows to $Q = 5{,}000$. Cancellation intensity rises to $\mu(5000) = 10$ per second — arrivals and executions haven't changed, but now cancellations dominate the mix, at $10/(8+10+3) = 47.6\%$ of the next event. A crowded queue empties itself from the back faster than a thin one, purely from cancellations, well before any trade happens.

Where it's used. These intensities are the building blocks of the The Queue-Reactive Model and of fill-probability estimates for resting orders: knowing the mix of rates at a level tells you the odds your own order gets executed before it gets overtaken by cancellations elsewhere in the queue. They're also the natural framework for point-process (Hawkes-type) models of Why Order Flow Is So Autocorrelated.